and F ds
9 Springer-Verlag 1989
The onset of deconfinement in SU(2) lattice gauge theory
J. Engels 1, J. Fingberg 1, K. Redlich 1'2, H. Satz 1'3, M. Weber 1 1 Fakultfit ffir Physik, Universit~it Bielefeld, D-4800 Bielefeld, F.R.G.
2 Institute for Theoretical Physics, University of Wroclaw, PL-50205 Wroclaw, Poland 3 Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Received 21 November 1988
Abstract. In SU(2) lattice gauge theory, we study de- viations from ideal gas behaviour near the deconfine- ment point. On lattices of size N~ x 4, N~ = 8, 12, 18 and 26, we calculate the quantity A - ( e - 3 P ) / T 4. It increases sharply just above T~, peaks at TITs= 1.15 -t-0.05 and then drops quickly. This form of behav- iour is shown to be the consequence of a second order phase transition. Dynamically it could arise because just above T~, the low m o m e n t u m states of the system are remnant massive modes rather than deconfined massless gluons.
1 Introduction
The thermodynamics of SU(2) gauge theory leads to a second order transition between a low temperature phase with gluonium constituents and a high temper- ature phase containing deconfined gluons. As a conse- quence of this transition, the energy density 8 of the system changes rapidly in a small interval around the critical temperature T= T~ ; it grows from a value of the size expected for an ideal gas of gluonium states to a much larger value close to the limiting form
F, st~/ T 4 -= 7c2/5 (1)
of an ideal gas of gluons with two spin and three colour degrees of freedom [1]. It was noted immedi- ately that the weak temperature dependence of e soon after T~ and its rapid convergence to values near esB do not imply the absence of interactions in the decon- fined phase. The quantity
A = (e -- 3 P)/T 4, (2)
where P denotes the pressure, vanishes for a massless ideal gas; for SU(2) gauge theory, it was found to differ from zero over a considerable range of tempera- tures above T~ [2]. Other lattice studies [3] have since
then c o n f r m e d this. Once the temperature is above the transition region, A decreases, and this decrease has been compared to the predictions of perturbation theory and of the bag model E4], as well as to a description including remnant massive modes at low momenta [3]. The functional behaviour of A has also been parametrized in terms of an ideal gas form for e, but with a pressure containing interaction effects linear in T [6, 7]; very recently, it was interpreted in terms of differences between chromoelectric and chromomagnetic contributions E3]. F r o m all these studies it has become increasingly clear that there are considerable thermodynamic interaction effects in the temperature range 1__< T/T~<=2--3. However, the de- tailed behaviour in the transition region and the ori- gin of the deviations from the ideal gas form (includ- ing the question of finite lattice size effects) has re- mained open. The aim of the present study is therefore to first calculate the relevant thermodynamic quanti- ties near T~ in a high statistics lattice evaluation, for a number of different spatial lattice sizes; we then show that the form of A we obtain is a consequence of the critical behaviour of the theory. Finally, we shall see that such behaviour can be understood dy- namically in terms of a rapid transition from massive to massless modes just above deconfinement [5, 8].
2 The lattice formulation
On an N ) x N~ lattice, the energy density ~ and the pressure P are determined by
e/T 4= 12N~4{g- 2(P~- P~)+G(Po--P~)+G(Po- P~)}, (3) p / T 4 = s / ( 3 T 4 ) - - 4 N r (P~+P~--2Po). (4) Here P~ and P~ denote the space-space and space- temperature plaquette averages, and Po that for a sym-
metric (N 4) lattice; g is the coupling constant. Asymp- totically, it is related to the lattice spacing a by
12~r 2 51 l l g 2 ) a A L = e x p
l l g 2 121 In .,,~j.~z~z~
(5)
The derivatives c~ and c~ have been evaluated in low- est order perturbation theory [9] giving
c~ = c o + cl, (6)
c, = -- c o + c2, (7)
with Co=0.1100325, c1=0.0040, Cz=0.04248 for the SU(2) case. Using (5), one has, again in lowest order,
dg -2
a d a - - 2 ( c 1 + c 2 ) ; (8)
this means that the entropy density s, s / T a = ( ~ + P ) / T 4 = 1 6 N 4 g - 2
[1 - g 2 ( c o + 1 / 2 q - 1 / 2 c z ) ] (P~- P0 (9) is determined completely in terms of the difference between space-space and space-time plaquettes. - F r o m (3), (4), we get
A=12N~ 4 a ~ a a d g - 2 [P~+P~-ZPo] (10) for the deviation A from an ideal gas of massless gluons.
3 L a t t i c e r e s u l t s
Our calculations were performed on N 3 x 4 lattices, with N~=8, 12 and 18; at a few temperature values, data were also taken with N~ = 26. In general, we used 100000 sweeps per temperature point; very close to T~, this was increased to 400000 sweeps per point.
The evaluation was carried out with a full group heat- bath vector program. F o r thermalization, the first 1000 iterations were discarded in data-taking.
With the same method, the symmetric plaquette values Po were calculated on a 164 lattice in the range 2 . 1 < 4 / g 2 < 2 . 9 5 at 3 0 p o i n t s with 1.4000-30000 sweeps. Part of these data are already published [10].
The results for e and P are shown in Figs. 1-2;
A is shown in Fig. 3. The data are compiled in Tables 1, 2 and 3. We note some finite size dependence near and just below T~, in particular for N~=8 and 12, and for e and hence A. On the whole, however, the results appear to stabilize with growing N~. In Fig. 4, we show A at T / A L = 4 2 . 3 ( T / T ~ ~ - 1) for different N~;
1.0
0 . 5 i
I ' I ' I '
0 0 0 0
O 0 0
O 0 []
o o o o o
D []
[]
D
0 8 o o o []
o o
6'D
0. Q t ~ [ I I I I I ,
40 60 80 lO0 T / A L
o E/ESB
[] P/PS8
Fig. 1. Energy density a n d pressure, calculated on the 183 x 4 lattice, normalized to the respective finite lattice Stefan-Boltzmann values [-12]
I . o E / E S B a n d P / P S B
0 . 5
I
26 <>
18 o o 12 x
8
Ab6
/
x" x ~
0. x I
30 40 50 T / A l
Fig. 2. Energy density a n d pressure, calculated on the 263 x 4, 183
• 4, 123 x 4 a n d 83 x 4 lattices and normalized as in Fig. 1 ; for the pressure only the points for 183 x 4 are s h o w n below T/AL=45
for N~>18, A seems to become quite constant. This is also seen when we compare the ratios of A for different N~ as a function of 4/g a. F o r T = T~, we can thus consider the results with N~ = 18 as at least indi- cative of the infinite volume limit.
t . 5
1 . 0
(Eps i l o n - 3 P ) / T ~ 4
<> 26 0 18 x 12
05 i o ~ o
O. , ~ t , t , I , I l
40 60 80 iO0 T/At.
Fig. 3. The interaction measure A for different lattice sizes
1.0
0 . 5 qATIO
' ' I ' ' ' ' I '
x
o o o o x
. . . .. . . . . .. . . . . . .. . . . . .. . . . . .. . " ~ . . . O -
0 o x x
<>
o
<>
o o x
D E L T A
I I I
X
o <>
o I I
1 0 20 N ~
0 . l l I , w l I I I i
2 . 3 0 2 . 3 5 4 / g z
Fig. 4. The ratios of A for N~=26 to N~=18 (O), 18 to 12 (o) and 12 to 8 ( x ) as function of 4/g z. In the inset, A is shown at the critical coupling 4/g2= 2.3 for different N~
The most striking feature is the pronounced sharp peak in A, which occurs at
T/T~-
1.15 _+0.05 (11)and thus is clearly above T~. We shall now show that this peak is associated to the occurence of a second order phase transition, and then study some possible dynamical mechanisms underlying this transition.
T a b l e 1. Numerical results from the 83 x 4 lattice 4/g 2 T e / r 4 P / T 4 s / T 3 A 2.2400 36.37 0.2567 0.0041 0.2608 (458) 2.2450 36.83 0.2892 0.0182 0.3073 (460) 2.2500 37.29 0.2896 - 0 . 0 0 1 4 0.2882 (458) 2.2550 37.76 0.3715 0.0187 0.3902 (460) 2.2600 38.24 0.4620 0.0196 0.4816 (459) 2.2650 38.72 0.4958 0.0116 0.5074 (459) 2.2700 39.21 0.5336 0.0068 0.5404 (459) 2.2750 39.70 0.6136 -0.0023 0.6113 (458) 2.2800 40.20 0.6638 0.0186 0.6824 (459) 2.2850 40.71 0.7753 0.0099 0.7852 (459) 2.2900 41.22 0.8327 0.0211 0.8538 (459) 2.2930 41.53 0.9020 0.0149 0.9169 (458) 2.2950 41.74 0.8903 0.0217 0.9120 (458) 2.2970 41.95 0.9316 0.0016 0.9332 (459) 2.2985 42.11 0.9893 0.0360 1.0253 (323) 2,3000 42.27 1.0400 0.0507 1.0907 (458) 2.3020 42.48 1.0781 0.0287 1.1067 (458) 2.3050 42.80 1.1102 0.0324 1.1426 (457) 2.3070 43.02 1.1503 0.0346 1.1849 (373) 2,3100 43.34 1.2060 0.0389 1.2449 (323) 2.3150 43.89 1.2464 0.0450 1.2914 (324) 2,3200 44.45 1.3822 0.0804 1.4625 (457) 2,3250 45.01 1.4300 0.0807 1.5107 (458) 2,3300 45.58 1.5164 0.0849 1.6014 (373) 2,3350 46.15 1.5388 0.0872 1.6260 (456) 2.3400 46.73 1.6626 0.1108 1.7734 (455) 2.3450 47.32 1.7134 0.1190 1.8324 (454) 2.3500 47.92 1.8277 0.1585 1.9862 (454) 2.3600 49.14 1.8918 0.1734 2.0652 (455) 2.3750 51.03 1.9475 0.2219 2.1694 (835) 2.4000 54.34 2.1767 0.3278 2.5045 (826) 2.4250 57.87 2.1277 0.3762 2.5039 (832) 2.4500 61.63 2.3244 0.5013 2.8256 (828) 2.4750 65.64 2.2262 0.5076 2.7338 (831) 2.5270 74.85 2.2852 0.5908 2.8759 (638) 2.6600 104.77 2.2659 0.6829 2.9489 (827)
0.2444 (443) 0.2346 (403) 0.2940 (414) 0.3153 (433) 0.4032 (404) 0.4608 (430) 0.5133 (469) 0.6205 (449) 0.6081 (441) 0.7458 (464) 0.7693 (458) 0.8573 (447) 0.8253 (466) 0.9269 (477) 0.8814 (382) 0.8878 (470) 0.9921 (468) 1.0129 (445) 1.0465 (407) 1.0893 (345) 1.1113 (340) 1.1411 (460) 1.1878 (451) 1.2616 (362) 1.2771 (397) 1.3302 (397) 1.3563 (340) 1.3520 (434) 1.3717 (379) 1.2818 (296) 1.1934 (285) 0.9990 (278) 0.8206 (271) 0.7035 (264) 0.5128 (389) 0.2171 (186)
4 I n t e r a c t i o n m e a s u r e a n d c r i t i c a l b e h a v i o u r
To study the effect of critical behaviour on the interac- tion measure A for T > T~, we consider a simple model obtained by modifying an ideal gas of massless con- stituents. The partition function
lnZ(T,V)=cVT3(1-t) 2-~, t<l
(12) witht-TdT,
constant c and critical exponent 0 < a<1, leads to an ideal gas for T ~ o o ; for T~T~, it results in singular behaviour.
F r o m (12) we obtain
P = c T4(1 - 0 2-~, (13)
e = 3 c T ~ [ ( 1 - t ) 2 - ~ + 1 / 3 ( 2 - c~) t(1 -- t)l-~], (14)
Table 2. Numerical results from the 123 x 4 lattice
4/g 2 T ~/T 4 P / T 4 s / T 3 A
2.1980 32.74 0.0929 0.0186 0.1115 (355) 0.0371 (355) 2.2600 38.24 0.1837 0.0204 0.2041 (250) 0.1225 (297) 2.2650 38.72 0.2507 0.0227 0.2735 (249) 0.1826 (285) 2.2700 3 9 . 2 1 0.2440 0.0123 0.2563 (250) 0.2071 (292) 2.2750 39.70 0.2843 0.0141 0.2984 (249) 0.2421 (289) 2.2800 40.20 0.3694 0.0199 0.3893 (250) 0.3097 (307) 2.2850 40.71 0.3779 0.0122 0.3901 (250) 0.3414 (345) 2.2880 41.02 0.4672 0.0171 0.4843 (249) 0.4159 (344) 2.2900 41.22 0.5162 0.0209 0.5371 (176) 0.4534 (244) 2.2930 41.53 0.6032 0.0225 0.6257 (176) 0.5358 (208) 2.2950 41.74 0.6149 0.0257 0.6406 (204) 0.5377 (316) 2.2970 41.95 0.6215 0.0229 0.6444 (250) 0.5530 (344) 2.2985 42.11 0.6934 0.0270 0.7204 (249) 0.6123 (314) 2.3000 42.27 0.7239 0.0291 0.7531 (203) 0,6365 (339) 2,3020 42.48 0.7897 0.0266 0.8163 (249) 0.7099 (407) 2.3050 42.80 0.8524 0.0228 0.8753 (249) 0.7839 (311) 2.3070 43.02 0.9175 0.0386 0.9561 (140) 0.8016 (279) 2.3100 43.34 0.9984 0.0380 1.0365 (204) 0.8844 (312) 2.3150 43.89 1 . 1 2 1 5 0.0545 1.1760 (249) 0.9580 (382) 2.3200 44.45 1.2614 0.0701 1,3315 (287) 1.0511 (320) 2.3250 45.01 1.3756 0.0885 1.4641 (249) 1.1102 (370) 2.3300 45.58 1.5122 0.0995 1,6118 (288) 1.2136 (323) 2.3400 46.73 1.6597 0.1244 1.7841 (287) 1.2863 (295) 2.3500 47.92 1.8032 0.1546 1.9577 (286) 1,3395 (302) 2.3620 49.39 1,9849 0.2122 2.1971 (452) 1,3483 (217) 2.3750 51.03 2,1119 0.2663 2.3783 (451) 1.3130 (241) 2.4000 54.34 2.1629 0,3471 2.5101 (453) 1.1215 (232) 2.4150 56.43 2.2521 0.4066 2.6587 (350) 1.0324 (300) 2.4250 57.87 2.2787 0.4303 2.7090 (452) 0.9879 (226) 2.4500 61.63 2.2959 0.4941 2.7900 (451) 0.8135 (222) 2.4750 65.64 2.3638 0.5679 2.9317 (452) 0.6601 (216) 2.5500 79.32 2.3688 0.6584 3.0272 (448) 0.3937 (155) 2.6600 104.77 2.4429 0.7557 3.1986 (451) 0.1759 (143)
for pressure and energy density; hence the specific heat ev=(Oe/~T)v diverges as ( l - t ) -" for t ~ l . F o r the interaction measure A we get
A = c ( 2 - ct) t(1 - 01 -~;
(15)
it vanishes for T--, T~ as well as for T-~ 0% and it has a p e a k at
T/T~ = t -1 = ( 2 - a). (16)
While the position of the peak is model-dependent (it is shifted by the addition of a non-singular term to In Z), the slope of A for T--* T~ has a universal form,
dA dT ~ ( 1 --t) -~, (17)
diverging at T = Tc with the same critical exponent as the specific heat.
The occurence of a p e a k in A just above T~ is thus a direct consequence of the second order decon-
Table 3. Numerical results from the 18 a x 4 lattice
4/g 2 T e/T'* P / T 4 s / T 3 d
2.1500 29.04 0.0316 0.0153 0.0469 (290) -0.0142 (268) 2.2700 39.21 0.1701 0.0244 0.1945 (191) 0.0969 (227) 2.2800 40.20 0.2181 0.0160 0.2341 (192) 0.1702 (219) 2.2850 40.71 0.3080 0.0227 0.3307 (191) 0.2400 (271) 2.2900 41.22 0.3572 0.0232 0.3804 (111) 0.2877 (281) 2.2950 41.74 0.4474 0.0235 0.4709 (136) 0.3769 (248) 2.2965 41.90 0.4754 0.0247 0.5002 (111) 0.4013 (183) 2.2970 41.95 0.5016 0.0266 0.5282 (93) 0.4217 (276) 2.2980 42.06 0.5258 0.0249 0.5507 (96) 0.4511 (201) 2.3000 42.27 0.5600 0.0258 0.5858. (90) 0.4826 (202) 2.3020 42.48 0.6579 0.0298 0.6877 (96) 0.5685 (214) 2.3050 42.80 0.7613 0.0371 0.7984 (111) 0.6499 (270) 2.3070 43.02 0.8267 0.0391 0.8659 (96) 0.7094 (190) 2.3100 43.34 0.9597 0.0458 1.0055 (111) 0.8222 (235) 2.3150 43.89 1.1208 0.0579 1.1787 (135) 0.9471 (320) 2.3170 44.11 1.1817 0.0623 1.2440 (111) 0.9949 (285) 2.3200 44.45 1.2896 0.0669 1.3566 (192) 1.0888 (332) 2.3300 45.58 1.5226 0.0938 1.6t64 (191) 1.2412 (261) 2.3400 46.73 1.7214 0.1329 1.8543 (191) 1.3226 (251) 2.3500 47.92 1.8666 0.1577 2.0243 (443) 1.3935 (160) 2.3630 49.51 1.9772 0.2010 2.1782 (346) 1.3741 (194) 2.3750 51.03 2.0719 0.2554 2.3272 (395) 1.3058 (205) 2.4000 54.34 2.2000 0.3541 2.5541 (468) 1.1378 (204) 2.4250 57.87 2.2841 0.4305 2.7146 (409) 0.9927 (195) 2.4500 61.63 2.2908 0.4992 2.7900 (656) 0.7932 (208) 2.4750 65.64 2.3412 0.5568 2.8980 (368) 0.6708 (184) 2.5300 75.41 2.3923 0.6475 3.0397 (905) 0.4498 (202) 2.5780 85.14 2.4306 0.7060 3.1366 (288) 0.3125 (170) 2.6210 94.92 2.4396 0.7398 3.1795 (294) 0.2201 (171) 2.6600 104.77 2.4643 0.7622 3.2266 (339) 0.1776 (115) Numerical results from the 263 x 4 lattice
4/g 2 T e / r ~ P / T 4 sIT 3 d
2.2750 39.70 0.1607 0.0270 0.1877 (390) 0.0796 (199) 2.2975 42.01 0.4276 0.0270 0.4545 (90) 0.3466 (273) 2.3000 42.27 0.5459 0.0295 0.5754 (157) 0.4574 (318)
finement trasition. Universality arguments [11] pre- dict for the a in SU(2) gauge theory the same value as in the three-dimensional Ising model, a-~ 0.11. This suggests that near T~, A should have the form
A =At(1-t)~ A, B = const. (18) In contrast to (15), we have also added a non-singular term B, since in the actual calculations e and P do not vanish for T = T~, as is the case for our simple m o d e l (12). W e w o u l d like to c o m p a r e this form with our data. T o do so, we have to take into a c c o u n t the difference between a c o n t i n u u m form and results obtained o n a finite lattice. These differences occur both because of the lattice cut-off [-12] and because g2:4:0 in the lattice evaluation. T o correct our mea- sured results for e and P, we divide the M o n t e Carlo values for these quantities by the corresponding w e a k
Table 4. Results from the 183 • 4 lattice corrected by weak coupling
4/g 2 T e/ewe P/Pwc A . . .
2.1500 29.04 0.0129 0.0186 --0.0112(21~
2.2700 39.21 0.0686 0.0294 0.0774 (181)
2.2800 40.20 0.0878 0.0192 0.1355 (174)
2.2850 40.71 0.1240 0.0272 0.1910 (216)
2.2900 41.22 0.1438 0.0278 0.2288 (223)
2.2950 41.74 0.1800 0.0282 0.2995 (197)
2.2965 41.90 0.1912 0.0297 0.3189 (146)
2.2970 41.95 0.2018 0.0320 0.3351 (219)
2.2980 42.06 0.2115 0.0299 0.3584 (159)
2.3000 42.27 0.2252 0.0310 0.3833 (160)
2.3020 42.48 0.2645 0.0358 0.4515 (170)
2.3050 42.80 0.3060 0.0445 0.5161(214)
2.3070 43.02 0.3323 0.0469 0.5632 (150)
2.3100 43.34 0.3856 0.0550 0.6526(186)
2.3150 43.89 0.4502 0.0694 0.7516 (254)
2.3170 44.11 0.4746 0.0747 0.7894 (226)
2.3200 44.45 0.5178 0.0802 0.8637 (263)
2.3300 45.58 0.6108 0.1123 0.9840 (207)
2.3400 46.73 0.6901 0.1591 1.0482 (198)
2.3500 47.92 0.7477 0.1886 1.1037 (127)
2.3630 49.51 0.7913 0.2402 1.0878(154)
2.3750 51.03 0.8284 0.3049 1.0335 (162)
2.4000 54.34 0.8781 0.4220 0.9004 (i6t)
2.4250 57.87 0.9101 0.5122 0.7854 (154)
2.4500 61.63 0.9111 0.5929 0.6281 (165)
2.4750 65.64 0.9296 0.6602 0.5317 (146)
2.5300 75.41 0.9465 0.7651 0.3580 (160)
2.5780 85.14 0.9588 0.8319 0.2505 (136)
2.6210 94.92 0.9598 0.8696 0.1782(138)
2.6600 104.77 0.9674 0.8939 0.1450 (94)
1.0
I 1 1
0 . 5 DELTA
0. I I I
42 4 3 4 4 45 TIAI.
Fig. 5. Tile behaviour of A just above T~, compared to the universal critical form, (18), with A=7.461, B=0.325
coupling values on the lattice [13]. The detailed form of the correction factors is given in the Appendix, and the corrected values of e, P, and A are listed in Table 4. The results for A are shown in Fig. 5 for temperatures just above T~, together with the form predicted by (18); the constants A and B are fitted.
We see that the lattice results indeed follow the ex- pected pattern.
5 T h e evolution o f gluon d e e o n f i n e m e n t
In this section, we want to consider a possible dynam- ical scheme leading to the observed behaviour of e, P and A. The decrease of A sufficiently far above Tc has been related to the presence of remnant massive modes by a n u m b e r of authors. Our aim here will be to account as well for the increase of A just above T~, and to understand how massive modes can lead to the calculated large deviations from ideal gas behaviour. Since both the energy density and the pres- sure of an ideal glueball system (with M-~ 1 GeV) are small compared to those of an ideal gluon gas, the mere presence of such modes above T~ cannot result in a A of the measured size. On the other hand, if the m o m e n t u m spectrum of the constituents contains massive modes at low m o m e n t a and massless gluons at high momenta, then the absence of low m o m e n t u m gluons, together with the temperature-dependence of the spectrum, results in a large and rapidly varying A, as we shall see.
To illustrate the effect, let us first consider an ideal Boltzmann gas of massless constituents, with a con- stant low m o m e n t u m cut-off K. Its partition function is
(3o
dV S dk kZ e-g/T,
In Z~ =~-~2 K (19)
and it leads to d K 3
A - 2zc2 T~ e - r / r (20)
This A has a peak at T = 1s vanishes as T -3 for large T and goes to zero exponentially as T ~ 0 . The reason for this behaviour is evident from the momen- tum spectrum in (19),
f ( k ) = k 2 e -k/r. (21)
It peaks at k = 2 T, so that for K > 2 T, the integral includes only the exponentially falling high-momen- tum part of the spectrum; with increasing T at fixed K (or with decreasing K), more and more of the full
spectrum is covered in the integration, and hence A approaches the ideal gas value.
While the overall functional form (20) is not unlike that found is Sect. 4, it yields a value of A which at the peak is much too small. Moreover, it contains no information about T~ and hence cannot reproduce the rapid variation in the critical region. To incorpo- rate the T~ dependence, we make the momentum cut- off K temperature-dependent. Since we expect no free gluons below T~, a natural form for the cut-off is
K/T~ = p
[ ( T - T~)/T~] -o; p, q = const. > 0. (22) With this cut-off, In ZG vanishes at T~, while at high T the full gluon spectrum is recovered.We now want to check if a description of this type, based on a rapid opening of the massless gluon sector just above T~, can indeed account for the ob- served behaviour of A. To obtain a full model, we make the ansatz
dv{i
l n Z - ~
d k k 2
l n ( 1 - e - k ~ v ~ / r )+ S dk k 2 ln(1--e-k/r)},
(23)K
with K given by (22). At T = T~, (23) reduces to a massive glueball gas; for T > T~, the massless gluons quickly take over. As degeneracy factor, we take d = 6;
this corresponds to two spin and three colour degrees of freedom for the gluons, and to degenerate 0+/2 + states for the glueballs. The critical temperature is Tc/AL=42, and we write the glueball mass
M=rT~.
The remaining three parameters p, q and r are deter- mined by fitting e, P and A obtained from (23) to the measured values listed in Table 2. The result is shown in Figs. 6 and 7 - we see that our simple model can indeed describe the behaviour quite well. One can certainly improve the agreement by constructing a model including further expected features. Bag pres- sure, perturbative corrections, colour corrections - these all will bring e down slightly from its ideal gas value and thus improve the fit. We have not done this, since any further terms with open parameters will obviously allow a good fit. Our essential point is to emphasize that the rapid growth of A just above T~ can be related to a rapid opening of the massless gluon sector; once the bulk of this sector is included, the corrections become smaller and A eventually goes to zero. The presence of massive modes near T~ is necessary to have non-vanishing thermodynamic quantities there, but its actual effect on the functional behaviour of the A obtained from (23) is quite small.
DELTA (corrected)
1 , 0
0 , 5
O,
f i
%
I ' I ' I
[ ]
I i
40 60 eO I00 T / A L
Fig. 6. The corrected interaction measure A ... compared to the phenomenological form obtained from (23) with M = 5.5 T~, p = 2.859 and q = 0.297
1.0/ ' I ~ [ ~ I ~ I '
P
0 0 0 0 00 rn.
[] []
0.5
0 .
40 60 80 i00 T / A L
Fig. 7. The same comparison as in Fig. 6, but for the corrected energy density and pressure
Acknowledgements. It is a pleasure to thank F. Karsch and B. Peters- son for many helpful discussions; to F. Karsch we are furthermore grateful for his program for the weak coupling expansions of the plaquettes. We are indebted to the H L R Z Jiilich and to the Bochum University computer centre for providing the necessary computer time.
Appendix A
T h e w e a k c o u p l i n g f o r m s o f the e n e r g y d e n s i t y a n d p r e s s u r e a r e o b t a i n e d f r o m (3)-(4) b y i n s e r t i n g the w e a k c o u p l i n g e x p a n s i o n s o f t h e p l a q u e t t e s . T h e y were c a l c u l a t e d u p to o r d e r g4 in [13] a n d h a v e t h e s t r u c t u r e
Pw~ = g2 N~ z - 1 p(z) + g4(N2 _ 1) p(4,) N~
(2N~ z - 3 ) ( N ~ : - 1)
q_ g4 Nc 2 p(4 b), (A. 1)
w h e r e N~ is the n u m b e r o f c o l o u r s a n d p ( 2 ) p ( 4 , ) a n d p(4b) are c o l o u r - i n d e p e n d e n t , b u t l a t t i c e size d e p e n - dent. D e n o t i n g t h e differences b e t w e e n p l a q u e t t e s b y d o u b l e i n d i c e s
P~p = P~ - Pp, (A.2)
o n e finds ( o n l y u p t o O (g2) b e c a u s e o f t h e f a c t o r g - 2) for % = 2
G, JT4=18N~a {P(Z~) +g 2
r"~D(4a)• D ( 2 ) • D ( 2 ) ' l t
(A.3)
Pw
2 T 4 = ~wd(3 T~) -- 8 Nr (cl + c2) g2 (Po(; / + ~o~'(:)~,. (A.4) F o r the w e a k c o u p l i n g c o r r e c t e d A o n e gets t h e nA . . . .
={e~"-3PP~"]/TL
(A.5)9 \ ewc Pwc//
o r
e P I T t 2
A ... = e~c P~,c] ~ - ' (A.6)
i.e., u p t o a c o n s t a n t factor, A ... is t h e difference o f the c o r r e c t e d C/esB a n d P/PsB values.
References
1. See e.g.H. Satz: Ann. Rev. Nucl. Part. Sci. 35 (1985) 245 2. J. Engels, F. Karsch, I. Montvay, H. Satz: Nucl. Phys. B205
[FS5] (1982) 545
3. V.V. Mitryushkin, A.M. Zadorozhny, G.M. Zinovjev: On ther- modynamic properties of chromoplasma, Dubna Preprint E2- 88-421 (1988)
4. H. Satz: Phys. Lett. l13B (1982) 245
5. F. Karsch: Z. Phys. C - Particles and Fields 38 (1988) 147 6. C.-G. K/illmann: Phys. Lett. 134B (1984) 363
7. M.I. Gorenstein, O.A. Mogilevsky: Z. Phys. C - Particles and Fields 38 (1988) 161
8. Hints for the presence of such modes were first presented in:
T.A. DeGrand, C.E. DeTar: Phys. Rev. D35 (1986) 742;
E. Manousakis, J. Polonyi: Phys. Rev. Lett. 58 (1987) 847;
J.B. Kogut, C.E. DeTar: Phys. Rev. D36 (1987) 2828 9. F. Karsch: Nucl. Phys. B205 [FS5] (1982) 285 10. J. Engels, J. Seixas: Phys. Lett. B206 (1988) 295
11. B. Svetitsky, L.G. Yaffe: Nucl. Phys. B210 [FS6] (1982) 423 12. J. Engels, F. Karsch, H. Satz: Nucl. Phys. B205 [FSS] (1982)
239
13. U. Heller, F. Karsch: Nucl. Phys. B251 [FS 13] (1985) 254
